Difference between revisions of "Core of a subgroup"
From Encyclopedia of Mathematics
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Let $H$ be a subgroup of $G$. The core of $H$ is the maximal subgroup of $H$ that is a [[normal subgroup]] of $G$. It follows that | Let $H$ be a subgroup of $G$. The core of $H$ is the maximal subgroup of $H$ that is a [[normal subgroup]] of $G$. It follows that | ||
$$ | $$ | ||
| − | \mathrm{core}_G (H) = \bigcap_g H^ | + | \mathrm{core}_G (H) = \bigcap_g H^g \ ,\ \ \ H^g = gHg^{-1} |
$$ | $$ | ||
If the index $[G:H] = n < \infty$, then $[G:\mathrm{core}_G (H)]$ divides $n!$. | If the index $[G:H] = n < \infty$, then $[G:\mathrm{core}_G (H)]$ divides $n!$. | ||
Latest revision as of 21:48, 30 November 2016
Let $H$ be a subgroup of $G$. The core of $H$ is the maximal subgroup of $H$ that is a normal subgroup of $G$. It follows that $$ \mathrm{core}_G (H) = \bigcap_g H^g \ ,\ \ \ H^g = gHg^{-1} $$ If the index $[G:H] = n < \infty$, then $[G:\mathrm{core}_G (H)]$ divides $n!$.
Let $g(xH) = (gx)H$ and define the permutation representation of $G$ on the set of right cosets of $H$ in $G$ (cf Coset in a group). Then its kernel is the core of $H$ in $G$.
References
| [a1] | M. Suzuki, "Group theory" , I , Springer (1982) |
| [a2] | W.R. Scott, "Group theory" , Dover, reprint (1987) (Original: Prentice-Hall, 1964) |
How to Cite This Entry:
Core of a subgroup. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Core_of_a_subgroup&oldid=33615
Core of a subgroup. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Core_of_a_subgroup&oldid=33615
This article was adapted from an original article by M. Hazewinkel (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article